STABILITY OF A PARAMETRICALLY EXCITED DAMPED INVERTED PENDULUM 1. INTRODUCTION

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1 Journa of Sound and Vibration (996) 98(5), STABILITY OF A PARAMETRICALLY EXCITED DAMPED INVERTED PENDULUM G. ERDOS AND T. SINGH Department of Mechanica and Aerospace Engineering, SUNY at Buffao, Buffao, NY 46, U.S.A. (Received 3 November 995, and in fina form 8 March 996). INTRODUCTION The transition of stabiity of the verticay upright position of the penduum from an unstabe to a stabe configuration has been iustrated in the iterature when the pivot of the penduum is subject to a periodic motion. Many studies, which have incuded anaytica and numerica approaches, have been used to determine the conditions for stabiity. The stabiity border has been cacuated for the undamped case by Landau and Lifshitz [] using the averaging method. Other anaytica toos, such as the method of sma parameters by Hsu [6], and the harmonic baance method by Cifford and Bishop [3], have been used to determine the stabiity border. A numerica approach was utiized by Capecchi and Bishop [4] to arrive at the stabiity border. Arnod [7] provides an eegant stabiity condition for uncompressibe fows using the trace of the Poincare map (or monodromy matrix). In this note we use the Foquet theory and variationa system to cacuate the stabiity border in cosed form. An overview of the mathematica toos that are used to cacuate the stabiity border are provided. We then compute the stabiity border in the viscous damped case, using the same toos as in the undamped case.. PARAMETRICALLY EXCITED UNDAMPED INVERTED PENDULUM In this section the stabiity of an undamped inverted penduum is considered. The pivot point of a simpe inverted penduum (Figure ) is subjected to vertica osciation of the form y(t)= cos ( t). Our goa here is to estabish the stabiity border that reates the ampitude ( ) to the frequency ( ) of the forcing function. The equation of motion of the inverted penduum is x = g cos ( t) sin (x), () where g is the acceeration due to gravity and is the ength of the penduum. Since we are interested in the stabiity of the verticay upright position of the penduum, we inearize equation () around x =, eading to 46X/96/ $5./ x = g 643 cos ( t) x. () 996 Academic Press Limited

2 644 LETTERS TO THE EDITOR Equation () can be represented in state space form as x g cos ( t) x = Equation (3) has the same form as the equation x x. (3) x = A(t)x, x R n, A(t + p)=a(t), (4) with a period p = /. The stabiity of the upright position depends on the characteristic roots of the period-advance map M = g p [7]. The characteristic roots and satisfy the characteristic equation of the matrix M, which are given by the equation where Tr(M) + Det (M)=, (5) Det (M)=e P Tr(A( )) d Tr (M)=Tr(g p ). (6,7) From equations (6), (7) and (3), it foows that Det (M)=e P d = : therefore The roots of equation (8) are Tr(M) +=. (8), = Tr (M) Tr (M) 4 (9) If abs (Tr (M)), then roots of equation (8) are rea and abs (Tr (M)) = abs ( + ). () Since the characteristic mutipiers of A(t) have to ie within the unit circe, we require, [7]. This requirement conficts with equation (), as the sum of the two roots is greater than. Therefore, for stabiity, we require The stabiity border is given by the equation abs (Tr (M)) = abs ( + ). () Tr (M) =. () Determination of the stabiity border requires a cosed form soution of equation (3), which is not avaiabe. However, if the ampitude of the excitation ( ) is sma, we can approximate the soution of equation (3) by expanding it into a Tayor series with respect m X ε cos (ωt) Figure. A verticay excited inverted penduum.

3 LETTERS TO THE EDITOR 645 to around =. Since the right side of equation (3) is a smooth function, the soution is differentiabe with respect to [9]. The soution (t) can be approximated as foows: (t, )=x(t,)+y(t,) + z(t,) +o( 3 ), (3) where x(t, )= (t, ), y(t,)=( (t, / )/ = and z(t, )= (t, )/ )/ =. To cacuate the M matrix we have to obtain the fundamenta matrix of equation (3). The coumn vectors of the fundamenta matrix are two ineary independent soutions of equation (3). Since equation (3) is inear, if the initia vaues are ineary independent, the corresponding soutions are ineary independent. We choose [, ] and [, ] as the set of initia vaues to cacuate two independent soutions, (t, ) and, (t, ) of equation (3): = A(t, ) where A(t, g )= cos ( t), () () =,. (4) This set of independent initia conditions resuts in the transition matrix (p, ) being equa to M. Substituting equation (3) into equation (4) and equating terms of equa powers of eads to the zeroth order equation x = A(t, )x, x() =,. (5) Soving equation (5), we obtain x, (t, ) and x, (t, ). The soution of the zeroth order equation appears as a forcing term in the first order equation, which can now be soved in cosed form. The first order equation is y = A (t,)y+ A(t, ) x i,j (t, ), y() =, i, j =,. (6) = Because the initia vaues of equation (4) do not depend on, the initia vaues of the variationa system are [, ]. The second order equation z = A (t,)z+ A(t, ) y i,j (t, ), z() =, i, j =,, (7) = can be soved in a simiar fashion. We now have a Tayor series soution which incudes terms up to the second order (equation (3)). The M period-advance map is given by the equation: M =, (p, ) (p, ), (p,, p = /. (8), (p, )

4 646. LETTERS TO THE EDITOR ε ω Figure. The stabiity border: ----, soution of Landau and Lifshitz;, soution by the proposed technique. Tr (M) is used to determine the stabiity border (equation ()), which eads to the equation, (p, )+, (p, ) =. (9) The cosed form soution of the inverted penduum was determined using a symboic manipuator, which resuted in the expression e g / +e g / + ( e4 g / ) 3 e g / g (4g + ) 3 =. () (a) 5 Penduum ange (rad) (b) Time (s) Figure 3. Numerica simuation: (a) =9, = 5; (b) =9, = 6. 3

5 LETTERS TO THE EDITOR 647 Area Area Tr M Det M Figure 4. Stabe areas for the damped case.. Soving equation () for, we have ( )= g /4 /4 +e g / (8g/ )+ 8e g / g +e g /. () Landau and Lifshitz [] soved for the stabiity border of an undamped inverted penduum using the average technique, which resuted in the equation g =. () ε c ω 8 6 Figure 5. The stabiity surface.

6 648 LETTERS TO THE EDITOR.4. ε ω Figure 6. Stabiity contours:, c =;, c =4; 3, c =6; 4, c =8. The stabiity borders resuting from equations () and (), which tend to merge with each other with increasing forcing frequency, are iustrated in Figure. Any combination of forcing ampitude and frequency that ies above the curves shoud resut in the verticay upright position of the penduum being a stabe equibrium position. For a forcing frequency of 9 rad/s, equation () requires the forcing ampitude to be greater than 5 for stabiity and equation () requires the forcing ampitude to be greater than 49. Thus, the two equations predict different characteristics for an forcing ampitude of 5. Numerica simuation of the non-inear undamped system for a forcing ampitude of 5 (Figure 3(a)), iustrates that the system is unstabe, exempifying that the stabiity border 5 4 (a) Penduum ange (rad) (b) Time (s) Figure 7. Numerica simuation: (a) =, = ; (b) =, = 3. 3

7 LETTERS TO THE EDITOR 649 estimated by the proposed approach is more accurate than that estimated by equation (). The expected stabiity of the equiibrium point for a forcing ampitude of 6 is confirmed in Figure 3(b). Figure 3(b) reveas the existence of a high frequency signa which corresponds to the frequency of excitation, superposed on a ow frequency signa. 3. PARAMETRICALLY EXCITED DAMPED INVERTED PENDULUM In this section we investigate the stabiity of the inverted penduum which is subject to viscous damping at the pivot. The dynamics of the damped inverted penduum are given by the state space equation x x = g cos ( t) c x x, (3) Equation (3) has the same form as equation (4) with a period p = /. The procedure to estabish the stabiity boundary is identica to the one used for the undamped system. The ony difference is that Tr (A), which resuts in constraints that are different from the undamped case to determine stabiity. From equations (6), (7) and (3) it foows that Det (M)=e P cd =e cp, where c is the damping coefficient and c. With the knowedge that the stabiity of the upright position of the penduum depends on the absoute vaues of the characteristic roots of the period-advance map M = g p, we substitute Det (M) into equation (5), resuting in the characteristic equation of the M matrix, the roots of which are Tr(M) +e cp =, (4), = Tr (M) Tr (M) 4 Det (M). (5) We investigate, separatey, the rea and the compex roots cases. It is evident that Tr (M) Det (M), C, Tr (M) Det (M), R. (6,7) In the compex case, Det (M)=e cp is smaer than, since c and p are positive numbers. This impies stabiity, since the product of the roots is ess than. In the rea case the necessary conditions for stabiity are satisfied if Tr (M) Tr (M) 4 Det (M), Tr (M)+ Tr (M) 4 Det (M). (8) Combining the stabiity constraints for the two cases, we have, from equations (6) and (8), e cp/ Tr (M) e cp/, Det (M) Tr (M) + Det (M), if Tr (M) Det (M) if Tr (M) Det (M). (9) From equation (9), the stabe area is given by the inequaity Tr (M) +e cp, (3)

8 65 LETTERS TO THE EDITOR which is iustrated in Figure 4. Area corresponds to equation (6) and area corresponds to equation (8). The cacuation of Tr (M) or, to be specific, the second order approximation of Tr (M), is carried out in the same fashion as the undamped case using variationa systems. The difference is the cacuation of the stabiity border, which is based on equation (3). Therefore, the stabiity border is a function of,, and c. In the interest of brevity, the cosed form expression representing the stabiity border is not incuded here. The surface that represents the stabiity border for the damped system is iustrated in Figure 5. Any combination of the damping constant, ampitude and frequency that ie above the surface eads to the upright position being a stabe equiibrium point. It can be seen from the graph that the stabiity is a strong function of the damping constant. Thus, for arger damping a arger ampitude of forcing is required to stabiize the system compared to a system with ow damping, for the same forcing frequency. The variation of the stabiity boundary for different vaues of the damping constant is demonstrated in Figure 6. This figure ceary shows that for given forcing frequency, the forcing ampitude increases with damping, for stabiity. The system response for a forcing frequency of rad/s and forcing ampitudes of and 3 is iustrated in Figure 7. The stabiity surface indicates that the forcing ampitude corresponding to a damping constant of 5 and a forcing frequency of is 34. For a forcing ampitude of, the system settes to the equiibrium position that corresponds to the verticay down postion and the forcing ampitude of 3 forces the system to come to rest at the verticay upright position. Thus, it is evident that the proposed approach to arrive at the stabiity surface is fairy accurate. 4. CONCLUSIONS A perturbation approach has been used to arrive at a cosed form soution of the stabiity surface for a damped inverted penduum. A recursive soution is used to arrive at the period-advance map, the eigenvaues of which determine the stabiity of the system. The stabiity border iustrates that, for high frequencies, the system is not a strong function of the damping in the system. However, at arger damping, the ampitude of forcing for a given frequency increases significanty, compared to the undamped case. Numerica simuations of the non-inear system are used to corroborate that the stabiity border predicted by the inearized approximation is reiabe. The soution for the undamped case, which is a specia case of the damped system, has been shown to cosey approach that determined by Landau and Lifshitz []. REFERENCES. A. STEPHENSON 98 Phiosophica Magazine 5, On induced stabiity.. L. D. LANDAU and E. M. LIFSHITZ 96 Mechanics. Oxford: Pergamon Press. 3. M. J. CLIFFORD and S. R. BISHOP 994 Journa of Sound and Vibration 74, Approximating the escape zone for the parametricay excited penduum. 4. D. CAPECCHI and S. R. BISHOP 994 Internationa Journa for the Dynamics and Stabiity of Systems 9, Periodic osciations and attracting basins for a parametricay excited penduum. 5. D. J. ACHESON 993 Proceedings of the Roya Society London 443, A penduum theorem. 6. C. S. HSU 975 Transactions of the American Society of Mechanica Engineers, Journa of Appied Mechanics 4, Limit cyce osciations of parametricay excited second-order noninear systems. 7. V. I. ARNOLD 973 Ordinary Differentia Equations. Transated and edited by Richard A. Siverman. Cambridge: MIT Press. 8. D. L. LUKES 98 Differentia Equations: Cassica to Controed. New York: Academic Press. 9. L. S. Pontriagin 96 Ordinary Differentia Equations Reading, MA: Addison-Wesey.